「Book Maker」

This ‘notebook’ consists of my run-through of Prof. Lorena Barba’s “12 steps to Navier-Stokes” course, because I admit I have a problem with fluids. The Navier-Stokes equations are: $$ \rho\frac{\partial \vec V}{\partial t} + \nabla \cdot \left(\rho \,\vec V \otimes \vec V + p\vec I \right) = \rho \vec g + \nabla \cdot \vec \tau + \vec f$$ The goal of this exercise to solve these equations numerically for some simple cases, starting from the ground-up. ... Read More
Meshing is a crucial process in obtaining accurate results for various simulations across different fields. In computational fluid dynamics, various meshing techniques are used in grid generation for 2D analyses of airfoils. Some nice run-throughs exist on YouTube, but they mostly deal with symmetric airfoils such as the beloved NACA 0012. I’ll attempt generating preliminary meshes over a cambered airfoil in this post and probably write a project once I’ve been able to perform flow analyses with accurate results. ... Read More
Bernard F. Schutz’s A First Course in General Relativity provides a nice introduction to the difficult subject in my opinion. In Chapter 6, he mentions that one should derive the Euler-Lagrange equations to minimise the spacetime interval of a particle’s trajectory, obtaining the geodesic equation: $$ \frac{\mathrm{d}}{\mathrm{d}\lambda}\left(\frac{\mathrm{d}x^{\gamma}}{\mathrm{d}\lambda}\right) + \Gamma^{\gamma}_{\;\alpha\beta}\frac{\mathrm{d}x^{\alpha}}{\mathrm{d}\lambda}\frac{\mathrm{d}x^{\beta}}{\mathrm{d}\lambda} = 0 $$ Note: At first I derived it from variational principles, but the Euler-Lagrange equations provide a faster route via means of a neat trick. ... Read More
I’ve been a rock/metal guitarist for the past 9 or 10 years, mostly concentrating on Western music and its theory; I am by no means an expert, but I’m curious about music theory and like to learn its different applications/interpretations in various styles. I had attended a Remember Shakti concert in 2012 which was my first real introduction to Indian classical musical elements. I know that it’s a fusion band, but I learnt the basic rhythmic elements during the concert thanks to my school’s guitar teacher, with whom I went to the concert. ... Read More
Learning about Lagrangian and Hamiltonian mechanics introduced me to an entirely new way of solving physics problems. The first time I’d read about this topic was in The Principle of Least Action chapter in Vol. 2 of The Feynman Lectures on Physics. I was introduced to a different perspective of viewing the physical world, perhaps a more general one than Newton’s laws. A famous example of a system whose equations of motion can be more easily attained using Lagrangian or Hamiltonian mechanics is the double pendulum. ... Read More
While reading through John D. Anderson Jr.’s derivation of minimum induced drag, I thought of a cool application of the calculus of variations in one of the equations to deduce the required condition. The equation that determines the downwash at a point is: $$w(y_0) = -\frac{1}{4\pi }\int^{b/2}_{-b/2} \frac{(\mathrm{d}\Gamma/\mathrm{d}y)}{y_0 - y}\mathrm{d}y = \int^{b/2}_{-b/2} \mathcal{L}(\Gamma,\Gamma’,y)\;\mathrm{d}y$$ This effectively implies that the downwash can be expressed as a functional of $\Gamma$, i.e. $w\left[\Gamma(y)\right]$, and one can find the functional derivative to find the extremal point. ... Read More
During my second year of the International Baccalaureate: Diploma Programme, we were assigned the task of making a portfolio for a problem statement in Mathematics HL. While I don’t have the problem statement anymore, I’ll outline the major points of the task: Find the solutions to $z^n - 1 = 0,\; z \in \mathbb{C}$. Plot these solutions on the Argand plane. Draw a tree diagram starting from the trivial solution $z = 1$ to every other root. ... Read More